{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:UKD7OKDA6S6H7ELDCRID4NBGGX","short_pith_number":"pith:UKD7OKDA","canonical_record":{"source":{"id":"1811.05538","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-11-13T21:52:53Z","cross_cats_sorted":[],"title_canon_sha256":"50cc447f9f2b9ebe3897bd3786fafad6c532de7b36583cd03f21727b986c66bc","abstract_canon_sha256":"4addba32f7a3616b42a79c81444a4ce51cbd940ec6167568671170dd3356b12e"},"schema_version":"1.0"},"canonical_sha256":"a287f72860f4bc7f916314503e342635deeb7d62c625e5dca294d21989fd184d","source":{"kind":"arxiv","id":"1811.05538","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.05538","created_at":"2026-05-17T23:48:22Z"},{"alias_kind":"arxiv_version","alias_value":"1811.05538v2","created_at":"2026-05-17T23:48:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.05538","created_at":"2026-05-17T23:48:22Z"},{"alias_kind":"pith_short_12","alias_value":"UKD7OKDA6S6H","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"UKD7OKDA6S6H7ELD","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"UKD7OKDA","created_at":"2026-05-18T12:32:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:UKD7OKDA6S6H7ELDCRID4NBGGX","target":"record","payload":{"canonical_record":{"source":{"id":"1811.05538","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-11-13T21:52:53Z","cross_cats_sorted":[],"title_canon_sha256":"50cc447f9f2b9ebe3897bd3786fafad6c532de7b36583cd03f21727b986c66bc","abstract_canon_sha256":"4addba32f7a3616b42a79c81444a4ce51cbd940ec6167568671170dd3356b12e"},"schema_version":"1.0"},"canonical_sha256":"a287f72860f4bc7f916314503e342635deeb7d62c625e5dca294d21989fd184d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:48:22.277576Z","signature_b64":"g6sW4ig0c8IeLZB9totquaEXwEtykbBy9JJAgwVzaak2Oy8tnDOc9zmiRnzilxoGNkMNUa9JDrQQymtkXUJECg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a287f72860f4bc7f916314503e342635deeb7d62c625e5dca294d21989fd184d","last_reissued_at":"2026-05-17T23:48:22.277059Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:48:22.277059Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1811.05538","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:48:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"QT9SzVHgerbXcTzYZDJxNk/9XM2NuyQee2WotdoKGlt20X1i85PIfYl086oz/5rbqtAaGKX/bVe1N4ILbHuzCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T22:08:48.696435Z"},"content_sha256":"d65bea5dbcd071bddd6de06454fa76dbe019ecc6bd3d94ee6c4b753699be8fea","schema_version":"1.0","event_id":"sha256:d65bea5dbcd071bddd6de06454fa76dbe019ecc6bd3d94ee6c4b753699be8fea"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:UKD7OKDA6S6H7ELDCRID4NBGGX","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Extensions of convex functions with prescribed subdifferentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.FA","authors_text":"Carlos Mudarra, Daniel Azagra, Javier G\\'omez-Gil, Juan Ferrera","submitted_at":"2018-11-13T21:52:53Z","abstract_excerpt":"Let $E$ be an arbitrary subset of a Banach space $X$, $f: E \\rightarrow \\mathbb{R}$ be a function, and $G:E \\rightrightarrows X^*$ be a set-valued mapping. We give necessary and sufficient conditions on $f, G$ for the existence of a continuous convex extension $F: X \\rightarrow \\mathbb{R} $ of $f$ such that the subdifferential $\\partial F$ of $F$ coincides with $G$ on $E.$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05538","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:48:22Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yQ3/iOo5F1jIZFL607y8wiJgTGR6gEDPes99JQklVolFBlBWYAUO7tuCBSifZCN20GuAiX/8zUZu/wZ4fu1TAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-05T22:08:48.697114Z"},"content_sha256":"f278883485d53695cba1593f1715718b2df8cd11c765398c07bbdc410e7d5208","schema_version":"1.0","event_id":"sha256:f278883485d53695cba1593f1715718b2df8cd11c765398c07bbdc410e7d5208"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UKD7OKDA6S6H7ELDCRID4NBGGX/bundle.json","state_url":"https://pith.science/pith/UKD7OKDA6S6H7ELDCRID4NBGGX/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UKD7OKDA6S6H7ELDCRID4NBGGX/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-05T22:08:48Z","links":{"resolver":"https://pith.science/pith/UKD7OKDA6S6H7ELDCRID4NBGGX","bundle":"https://pith.science/pith/UKD7OKDA6S6H7ELDCRID4NBGGX/bundle.json","state":"https://pith.science/pith/UKD7OKDA6S6H7ELDCRID4NBGGX/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UKD7OKDA6S6H7ELDCRID4NBGGX/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:UKD7OKDA6S6H7ELDCRID4NBGGX","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4addba32f7a3616b42a79c81444a4ce51cbd940ec6167568671170dd3356b12e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-11-13T21:52:53Z","title_canon_sha256":"50cc447f9f2b9ebe3897bd3786fafad6c532de7b36583cd03f21727b986c66bc"},"schema_version":"1.0","source":{"id":"1811.05538","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1811.05538","created_at":"2026-05-17T23:48:22Z"},{"alias_kind":"arxiv_version","alias_value":"1811.05538v2","created_at":"2026-05-17T23:48:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1811.05538","created_at":"2026-05-17T23:48:22Z"},{"alias_kind":"pith_short_12","alias_value":"UKD7OKDA6S6H","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"UKD7OKDA6S6H7ELD","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"UKD7OKDA","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:f278883485d53695cba1593f1715718b2df8cd11c765398c07bbdc410e7d5208","target":"graph","created_at":"2026-05-17T23:48:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"Let $E$ be an arbitrary subset of a Banach space $X$, $f: E \\rightarrow \\mathbb{R}$ be a function, and $G:E \\rightrightarrows X^*$ be a set-valued mapping. We give necessary and sufficient conditions on $f, G$ for the existence of a continuous convex extension $F: X \\rightarrow \\mathbb{R} $ of $f$ such that the subdifferential $\\partial F$ of $F$ coincides with $G$ on $E.$","authors_text":"Carlos Mudarra, Daniel Azagra, Javier G\\'omez-Gil, Juan Ferrera","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-11-13T21:52:53Z","title":"Extensions of convex functions with prescribed subdifferentials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1811.05538","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:d65bea5dbcd071bddd6de06454fa76dbe019ecc6bd3d94ee6c4b753699be8fea","target":"record","created_at":"2026-05-17T23:48:22Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4addba32f7a3616b42a79c81444a4ce51cbd940ec6167568671170dd3356b12e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2018-11-13T21:52:53Z","title_canon_sha256":"50cc447f9f2b9ebe3897bd3786fafad6c532de7b36583cd03f21727b986c66bc"},"schema_version":"1.0","source":{"id":"1811.05538","kind":"arxiv","version":2}},"canonical_sha256":"a287f72860f4bc7f916314503e342635deeb7d62c625e5dca294d21989fd184d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a287f72860f4bc7f916314503e342635deeb7d62c625e5dca294d21989fd184d","first_computed_at":"2026-05-17T23:48:22.277059Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:48:22.277059Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g6sW4ig0c8IeLZB9totquaEXwEtykbBy9JJAgwVzaak2Oy8tnDOc9zmiRnzilxoGNkMNUa9JDrQQymtkXUJECg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:48:22.277576Z","signed_message":"canonical_sha256_bytes"},"source_id":"1811.05538","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d65bea5dbcd071bddd6de06454fa76dbe019ecc6bd3d94ee6c4b753699be8fea","sha256:f278883485d53695cba1593f1715718b2df8cd11c765398c07bbdc410e7d5208"],"state_sha256":"043ee89e5911b73ae1c0c5d0bb3c8b38788c380d7ef9eb3fb9f51aac5e8334f6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6qWrGr0m3a61yAZPOC2/gfODmHUjFqYDYtaqe7M2pkobqIlSCHPzdBb3IYywcGCJdoETAFmZwOGf/5ygqnZfAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-05T22:08:48.700525Z","bundle_sha256":"ba1349fcef26def7b6eb28449ba208ca613ae4ba7229ffb8b47ae7c1d301c6f6"}}